naginterfaces.library.linsys.real_​posdef_​band_​solve

naginterfaces.library.linsys.real_posdef_band_solve(uplo, kd, ab, b)

real_posdef_band_solve computes the solution to a real system of linear equations $AX=B$, where $A$ is an $n\times n$ symmetric positive definite band matrix of band width $2k+1$, and $X$ and $B$ are $n\times r$ matrices. An estimate of the condition number of $A$ and an error bound for the computed solution are also returned.

For full information please refer to the NAG Library document for f04bf

https://support.nag.com/numeric/nl/nagdoc_30.3/flhtml/f04/f04bff.html

Parameters

uplostr, length 1

If $uplo=\text{'U'}$, the upper triangle of the matrix $A$ is stored.

If $uplo=\text{'L'}$, the lower triangle of the matrix $A$ is stored.

kdint

The number of superdiagonals $k$ (and the number of subdiagonals) of the band matrix $A$.

abfloat, array-like, shape $(kd+1,n)$

The $n\times n$ symmetric band matrix $A$. The upper or lower triangular part of the symmetric matrix is stored in the first $kd+1$ rows of the array. The $j$th column of $A$ is stored in the $j$th column of the array $ab$ as follows:

See Further Comments below for further details.

bfloat, array-like, shape $(n,\text{nrhs})$

The $n\times r$ matrix of right-hand sides $B$.

Returns

abfloat, ndarray, shape $(kd+1,n)$

If the function exits successfully or $errno$ = $n$ + 1, the factor $U$ or $L$ from the Cholesky factorization $A=U^{T}U$ or $A=L{L}^T$, in the same storage format as $A$.

bfloat, ndarray, shape $(n,\text{nrhs})$

If the function exits successfully or $errno$ = $n$ + 1, the $n\times r$ solution matrix $X$.

rcondfloat

If the function exits successfully or $errno$ = $n$ + 1, an estimate of the reciprocal of the condition number of the matrix $A$, computed as $rcond=1/\left({\parallel A\parallel }_1{\Vert A^{-1}\Vert }_1\right)$.

errbndfloat

If the function exits successfully or $errno$ = $n$ + 1, an estimate of the forward error bound for a computed solution $\hat{x}$, such that ${\parallel \hat{x}-x\parallel }_1/{\parallel x\parallel }_1\le errbnd$, where $\hat{x}$ is a column of the computed solution returned in the array $b$ and $x$ is the corresponding column of the exact solution $X$. If $rcond$ is less than machine precision, $errbnd$ is returned as unity.

Raises

NagValueError

(errno $-4$)

On entry, $\text{nrhs}=⟨value⟩$.

Constraint: $\text{nrhs}\ge 0$.

(errno $-3$)

On entry, $kd=⟨value⟩$.

Constraint: $kd\ge 0$.

(errno $-2$)

On entry, $n=⟨value⟩$.

Constraint: $n\ge 0$.

(errno $-1$)

On entry, $uplo$ not one of ‘U’ or ‘u’ or ‘L’ or ‘l’: $uplo=⟨value⟩$.

(errno $(i>0)\text{and}(i\le n)$)

The principal minor of order $⟨value⟩$ of the matrix $A$ is not positive definite. The factorization has not been completed and the solution could not be computed.

Warns

NagAlgorithmicWarning

(errno $n+1$)

A solution has been computed, but $rcond$ is less than machine precision so that the matrix $A$ is numerically singular.

Notes

The Cholesky factorization is used to factor $A$ as $A=U^{T}U$, if $uplo=\text{'U'}$, or $A=L{L}^T$, if $uplo=\text{'L'}$, where $U$ is an upper triangular band matrix with $k$ superdiagonals, and $L$ is a lower triangular band matrix with $k$ subdiagonals. The factored form of $A$ is then used to solve the system of equations $AX=B$.

References

Anderson, E, Bai, Z, Bischof, C, Blackford, S, Demmel, J, Dongarra, J J, Du Croz, J J, Greenbaum, A, Hammarling, S, McKenney, A and Sorensen, D, 1999, LAPACK Users’ Guide, (3rd Edition), SIAM, Philadelphia, https://www.netlib.org/lapack/lug

Higham, N J, 2002, Accuracy and Stability of Numerical Algorithms, (2nd Edition), SIAM, Philadelphia